Fraser and Schoen have uncovered a beautiful relationship between free boundary minimal surfaces in the unit ball and the Steklov problem: the coordinate functions of such surfaces are Steklov eigenfunctions with eigenvalue 1, and, on the other hand, the eigenfunctions for extremal metrics for the Steklov problem provide embeddings of free boundary minimal surfaces. The Fraser–Li conjecture states that not only are the coordinate functions Steklov eigenfunctions with eigenvalue 1, this eigenvalue is also the smallest non-zero one.

In this talk, I will discuss the history of the problem, the relation with minimal surfaces in the sphere, and explain an elementary proof of special cases of the Fraser–Li conjecture assuming some additional symmetries.